Proof (Intuition) Examples: 46 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Proof (Intuition).

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The informal, intuitive sense of why a mathematical statement must be true — the "aha" that precedes and motivates a formal proof.

A chain of reasoning that convinces you something MUST be true.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Proof intuition is the convincing 'aha' chain that something cannot fail to be true, which then guides a formal proof.

Common stuck point: The procedure for proof (intuition) is the easy part; the trap is accepting a pile of confirming examples as the intuition. Asking "Do I have a chain of reasoning that forces the conclusion, beyond just examples that happen to work?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Do I have a chain of reasoning that forces the conclusion, beyond just examples that happen to work?

Worked Examples

Example 1

easy

Before writing a formal proof that 'the sum of two even integers is even,' build the intuition. Explain why it must be true, then formalise.

Answer

a + b = 2 (m + n) is even

First step

Intuition: Even numbers are multiples of 2. Adding two multiples of 2 gives another multiple of 2 — the '2-ness' is preserved.

Full solution

2
Analogy: pairs of objects combined with pairs of objects always give pairs.
3
Formalise: Let $a=2m$ and $b=2n$ . Then $a+b=2m+2n=2(m+n)$ , which is even.

Proof intuition means building a convincing internal picture before writing the formal argument. The intuition guides which definitions and algebraic steps to use, making the formal proof feel natural rather than mechanical.

Example 2

medium

Build intuition for why

\sqrt{2}

is irrational before writing the formal proof. What is the core contradiction?

Example 3

medium

Build the intuition for why the sum of an odd and an even integer is odd. Then prove it formally.

Example 4

medium

Build intuition for why the product of any three consecutive integers is divisible by

6

, then prove it.

Example 5

medium

Build intuition for why there are infinitely many primes (Euclid). Sketch the key idea.

Example 6

hard

Build intuition for the contradiction at the heart of proving

\sqrt 3

is irrational. Sketch the argument.

Example 7

hard

Why does the principle of strong induction allow assuming

P(1), P(2), \ldots, P(k)

all at once when proving

P(k+1)

? Build intuition with the prime factorization theorem.

Example 8

hard

Sketch a proof by contradiction that there is no rational number whose square is

5

Example 9

challenge

Sketch a proof using the well-ordering principle that every positive integer has a unique prime factorization.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Build intuition for the statement: 'For any integer

n

n(n+1)

is even.' Explain informally why this must be true.

Example 2

medium

Build intuition for induction: why does proving '

P(k) \Rightarrow P(k+1)

' together with

P(1)

establish

P(n)

for all

n

Example 3

easy

Does checking that

3,5,7

are odd prove 'all primes are odd'? Give

1

for yes,

0

for no.

Example 4

easy

To disprove 'all swans are white', how many counterexamples suffice? Give the number.

Example 5

easy

The sum of two even numbers is even. Writing

2a+2b=2(a+b)

, what common factor proves it? Give the factor.

Example 6

easy

In a proof by contradiction of '

\sqrt2

is irrational', we assume

\sqrt2=\frac{p}{q}

in lowest terms. What parity does

p

turn out to share with

p^2

here? Give 'even' as

1

Example 7

easy

A proof must state assumptions. In 'if

n

is even then

n^2

is even', what is assumed about

n

? Give 'even' as

1

Example 8

easy

Does the converse of 'if it rains, the ground is wet' (i.e. 'if wet then rains') follow automatically? Give

1

for yes.

Example 9

easy

How many cases does a proof by cases on the parity of an integer need? Give the number.

Example 10

easy

The contrapositive of 'if

P

then

Q

' is 'if not

Q

then not

P

' and is logically equivalent. Give

1

if equivalent.

Example 11

medium

Induction proves

P(n)

for all

n\ge1

via base case and inductive step. For

\sum_{i=1}^{n} i=\frac{n(n+1)}{2}

, what is the base case value at

n=1

Example 12

medium

In the inductive step for

\sum_{i=1}^{n} i=\frac{n(n+1)}{2}

, assuming it for

n=k

, what do you add to both sides to reach

n=k+1

? Give the term.

Example 13

medium

A pigeonhole proof: placing

13

people into

12

months guarantees at least how many share a month?

Example 14

medium

To prove 'the product

n(n+1)

is always even', which property of two consecutive integers is the key insight? Give 'one is even' as the count of even factors guaranteed.

Example 15

medium

A valid proof needs each step to follow. In '

a=b

, so

a^2=ab

', what operation was applied to both sides? Give the multiplier.

Example 16

medium

The infinitude-of-primes proof multiplies known primes and adds

1

. For primes

\{2,3\}

, what is

2\cdot3+1

Example 17

challenge

In a flawed proof '

1=2

', both sides are divided by

(a-b)

after setting

a=b

. What is the value of

a-b

that invalidates this step?

Example 18

challenge

A proof shows

n^3-n

is divisible by

3

for all integers

n

. Factoring gives

(n-1)n(n+1)

— how many consecutive integers is that product, guaranteeing a multiple of

3

Example 19

challenge

Why must a proof's logic be valid even if the conclusion is true? If '

2+2=4

' is justified by 'because the sky is blue', is the reasoning valid? Give

1

for valid.

Example 20

medium

To prove a number is divisible by

6

, it suffices to show divisibility by which two coprime numbers? Give their product.

Example 21

medium

A direct proof that

n

even implies

n^2

even writes

n=2k

. What is

n^2

in terms of

k

(give the coefficient of

k^2

Example 22

medium

In an 'if and only if' proof, how many directions must be shown? Give the number.

Example 23

easy

How many counterexamples are needed to disprove a universal statement of the form 'for all

x

P(x)

Example 24

easy

Is checking that

4, 6, 8

are even sufficient to prove 'all even numbers

\ge 4

are even'?

Example 25

easy

True or false: 'if

P

then

Q

' is logically equivalent to its contrapositive 'if not

Q

then not

P

Example 26

easy

Is the statement 'for some

n

n^2 = n

' true? Give an example.

Example 27

easy

The proof technique that assumes the opposite and derives a contradiction is called proof by _____.

Example 28

medium

Sketch the intuition behind the pigeonhole principle: if

n + 1

pigeons fit into

n

holes, what must happen?

Example 29

medium

Give a counterexample to the claim 'every odd integer is prime'.

Example 30

medium

Why is proving

P \to Q

by contrapositive sometimes easier than direct proof? Illustrate with: 'if

n^2

is even, then

n

is even'.

Example 31

medium

Give the contrapositive of 'if

n

is divisible by

6

, then

n

is divisible by

2

Example 32

medium

Why does '

P

implies

Q

' NOT mean '

Q

implies

P

'? Give a real-world example.

Example 33

medium

Disprove 'every positive integer is the sum of two squares' with a counterexample.

Example 34

hard

Use the pigeonhole principle to show that among any

13

people, at least two share a birth month.

Example 35

hard

Why does the statement '

P

if and only if

Q

' require TWO proofs?

Example 36

hard

Using induction, sketch the proof that

1 + 2 + \cdots + n = n(n+1)/2

for all

n \ge 1

Example 37

challenge

Prove or disprove: 'There exist irrational numbers

a, b

such that

a^b

is rational.'

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Related Concepts

Logical Statement Conditional Statement Contrapositive Counterexample

Background Knowledge

These ideas may be useful before you work through the harder examples.

logical statementconditional